Gompertz regression parameterized as accelerated failure time model Filip Andersson and Nicola Orsini Biostatistics Team Departmentof Public Health Sciences Karolinska Institutet 2017 Nordic and Baltic Stata meeting
Content § Introduction § Proportional hazard model § Accelerated failure time model § The Gompertz distribution § Structural equation models and mediation § Mediation in survival models § Estimating confidence intervals § What I am working on Filip Andersson 2017-08-31 2
Content § Example à Data à Pre-estimation à Gompertz proportional hazard à Cox regression à Gompertz vs. Kaplan-Maier à Gompertz ATF model à Post-estimation à Conclusion Filip Andersson 2017-08-31 3
Introduction § Why use parametric surival models? à Can handle right-, left- or interval-censored data à Cox regression can’t handle left- or interval-censored data à Produce better estimation if you have a theoretical expectation of the baseline hazard à Can estimate expected life, not only hazard ratios (AFT-models) à Can include random effects – frailty models (not discussed here) Filip Andersson 2017-08-31 4
Introduction § A model that is lacking an easy way to estimate in Stata à Gompertz regression parameterized as accelerated failure time model à Exist in R § eha-package, with command: aftreg § Why use Stata? à Easy handling survival data § Data management § Setup à Good graphical possibility Filip Andersson 2017-08-31 5
Proportional hazard model § Easy to compare with Cox regression à Hazard ratios à Plots § Cummulative hazard function § Survival function à Commonly used § Hazard function general form à ℎ 𝑢 𝑦 = ℎ % (𝑢)𝑓 )* Filip Andersson 2017-08-31 6
Accelerated failure time model § Can be seen as a linear model (simplest form): à log 𝑢 = 𝑏 + 𝑐𝑦 + 𝜁 à Usefulin mediation § Estimation on life scale à Estimation of expected baseline life § Area under the survival curve when all covariates are zero à Compare expected life between two groups § Logarithmic change in expected life compared to the baseline life expectancy § Expected life = Baseline life expectancy ∗ exp (effect) Filip Andersson 2017-08-31 7
Accelerated failure time model § Definiton of accelerated failure time model à For a group (X 1 ,X 2 …X p ) , the model is written mathematically as C 𝑇 𝑢 𝑦 = 𝑇 % D()) , where S 0 (t) is the baseline survival function and 𝜃(𝑦) is an acceleration factor that is a ratio of survival times corresponding to any fixed value of S(t). The acceleration factor is given according to the formula 𝜃 𝑦 = 𝑓 (F G ) G H⋯HF J ) J ) . (Qi, J (2009)) § Hazard function K C à ℎ 𝑢 𝑦 = D()) ℎ % D()) § Log-linear from à log 𝑢 = 𝑏 + 𝑐𝑦 + 𝜏𝜁 à Where t and ε following corresponing distributions Filip Andersson 2017-08-31 8
The Gompertz distribution § When is it useful? à Adult and old age mortality for humans § Demographic models § Including models with treatment effects, such as cancer patiens § Can be problem with very old individuals § Normal paramertization à ℎ 𝑢 = 𝜇𝑓 NC à 𝜇 > 0, 𝛿 ≥ 0, 𝑢 > 0 Filip Andersson 2017-08-31 9
The Gompertz distribution § Suggested new parametrization by Broström, G & Edvinsson, S (2013) à 𝜇 → U N , 𝛿 → K N à ℎ 𝑢 = U V W X N 𝑓 à 𝜇 > 0, 𝛿 > 0, 𝑢 > 0 § Proof of new parametrization à Hazard for AFT-model K C à ℎ 𝑢 𝑦 = D()) ℎ % D()) à Here, new gamma can be seen as an accelerated factor Filip Andersson 2017-08-31 10
The Gompertz distribution log 𝑢 = 𝑏 + 𝑐𝑦 + 𝜁 § Linear model: § Here, ε is following a log-Gompertz or inverse Weibull distribution § Compare to the Weibull model, where ε follows a Gumbel distribution Likelihood function § X − 1 V W 𝑇 𝑢 = 𝑓𝑦𝑞 −𝜇 𝑓 à Survivalfunction: 𝐺 𝑢 = ℎ 𝑢 𝑇 𝑢 à Density function: ℎ 𝑢 = U V W X N 𝑓 à Hazard function: ℎ a 𝑢 a 𝑇 a (𝑢 a ) b c 𝑇 a (𝑢 a ) Kdb c e à 𝑀 𝛽, 𝜈, 𝜏 = ∏ afK Filip Andersson 2017-08-31 11
Structural equation models and mediation § Simple way to estimate linear models within a pathway framework § Estimate all equations and combine for the direct and indirect effects § Supported by most statistical programs à In Stata the gsem-command combined with simulation is preferable Filip Andersson 2017-08-31 12
Mediation in survival models § What do we need to do? 1. Estimate a parametric survivalmodel 2. Estimate the exposure on the mediator § First two steps directly from the gsem output 3. Estimate the indirect, direct and total effect 4. Estimate confidence intervals and significance § Step three and four can be done with either simulation or delta method § These models are simple for continous mediators, but can be tricky with binary or categorical mediators Filip Andersson 2017-08-31 13
Estimating confidence intervals § Simulation à Boostraping § Seems to be the more popular simulation method § Calculate point estimates for the indirect and direct effects § Simulate these point estimates à Monte carlo simulation § More flexible to handle problematic correlations § Not as straight forward § Delta method § Easiest method and probably most popular § Need a stronger assumption of normality Filip Andersson 2017-08-31 14
What I am working on § A Stata command, staftgomp , to estimate the Gompertz regression parameterized as accelerated failure time model similar to what streg does § A post-estimation command that would make it simple to estimate direct, indirect and total effect, with confidence intervals, for survival models Filip Andersson 2017-08-31 15
Example § Scanian Economic Demographic Database (Bengtsson, T., Dribe, M. and Svensson, P. (2012)) § Longitudinal historical database à Data from 17 th century and onwards à Here, data from individuals born between 1815-1860 are used à Comes from five rural parishes in western Scania à Consist of important life course events as birth and death, but also births of children, marriage or socioeconomic status are recorded Filip Andersson 2017-08-31 16
Data § Variables used: à ”Treatmentvariable”: § Approximation of bad early life conditions § Infant mortality rate at the year of birth § High imr vs. low imr (binary) § Years of high diseaseload such as measles, smallpox and whooping cough (Quaranta, L. (2013)) à Parentalsocioeconomic status § Socioceconomic status at birth (binary) § Confounder à Outcome § The individuals are followed until death or out-migration. Filip Andersson 2017-08-31 17
Pre-estimation § Compare hazard estimations of Gompertz proportional hazard model and Cox regression § Plot survival curve and compare with Kaplan-Maier § If not acceptable test with different survival distribution until the parametric model is acceptable à Here, we choose Gompertz as it fits good and are supported theoretically for adult mortality Filip Andersson 2017-08-31 18
Gompertz proportional hazard . streg imr_high ses, dist(gompertz) Gompertz regression -- log relative-hazard form No. of subjects = 3,756 Number of obs = 3,756 No. of failures = 880 Time at risk = 19824107 LR chi2(2) = 26.53 Log likelihood = -1773.9194 Prob > chi2 = 0.0000 ------------------------------------------------------------------------------ _t | Haz. Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- imr_high | 1.259023 .0951873 3.05 0.002 1.085624 1.460119 ses | 1.362878 .1010669 4.17 0.000 1.178513 1.576084 _cons | 9.57e-06 8.25e-07 -134.05 0.000 8.08e-06 .0000113 -------------+---------------------------------------------------------------- /gamma | .0002332 8.35e-06 27.92 0.000 .0002168 .0002496 ------------------------------------------------------------------------------ Filip Andersson 2017-08-31 19
Cox regression . stcox imr_high ses Cox regression -- Breslow method for ties No. of subjects = 3,756 Number of obs = 3,756 No. of failures = 880 Time at risk = 19824107 LR chi2(2) = 28.17 Log likelihood = -5889.8259 Prob > chi2 = 0.0000 ------------------------------------------------------------------------------ _t | Haz. Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- imr_high | 1.261686 .0955679 3.07 0.002 1.087617 1.463614 ses | 1.381581 .102833 4.34 0.000 1.194043 1.598573 ------------------------------------------------------------------------------ Filip Andersson 2017-08-31 20
Gompertz vs. Kaplan-Maier Filip Andersson 2017-08-31 21
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